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In some geometries, parallel lines "meet/touch/coincide" at infinity. This being the case, there must necessarily be an angle between them. I was wondering what the "value" of this angle would be. Is it always $\pi/2$? Is it $0$? Is it infinite? is it $2\pi$? Or is there some formula which makes the angle variable depending on the perpendicular distance between the lines?

I'm particularly interested in answers that approach the question from multiple different geometries, including geometries where parallel lines don't meet (in which case the question becomes, "what is the angle between two lines which don't meet?"). As mentioned, the concept of "angle" is meaningless in projective geometry. What does this question look like from the perspective of hyperbolic, euclidean, and elliptical geometries?

(It has been a while since I've done serious mathematics and my terminology might be off. I've put words which I'm not sure about in scare quotes. Feel free to edit.)

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    $\begingroup$ I believe the angle would be $0$. The stereographic projection preserves angles (or so I've heard), and the circles on the sphere that represent two parallel lines are tangent to each other at the point at infinity, so they seem to meet at angle $0$. $\endgroup$
    – Polygon
    Jul 15, 2020 at 2:35
  • $\begingroup$ You can use limits to demonstrate that it is 0. draw a triangle and keep two of the points a fixed distance apart while dragging them downwards. The angle at the other point will approach 0 $\endgroup$ Jul 15, 2020 at 3:21
  • $\begingroup$ It seems to me that it's in Euclidean geometry that parallel lines can be said to meet at infinity (although it is a bit of an abuse of language to say so). In Lobachevskian geometry, parallel lines generally never meet, at infinity or anywhere else. In spherical geometry, there are no parallel lines, and no infinity. So just which geometries do you have in mind, TheIron? $\endgroup$ Jul 15, 2020 at 4:06
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    $\begingroup$ In hyperbolic geometry, convergently parallel lines make an angle of $0$ (in a limiting sense). Note that the area of a triangle in the hyperbolic plane (of curvature $-1$) is given by the "angular defect", the amount the angle-sum falls short of $\pi$; ie, $\pi-(\text{angle sum})$. In a triangle whose three vertices are "ideal" points at infinity (ie, one with sides are pair-wise convergently parallel), the angles are $0$, so that the angular defect —and thus the area— is $\pi$, making that value the largest possible area for a hyperbolic triangle. Pretty neat, that. $\endgroup$
    – Blue
    Jul 15, 2020 at 6:02
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    $\begingroup$ It's perhaps worth noting that in, say, the Poincaré Disk Model of hyperbolic geometry, lines are represented by arcs of circles orthogonal to the "line (circle) at infinity". Moreover, the model is "conformal": angles between the (tangents to) the Euclidean arcs accurately reflect the angles between the hyperbolic lines they represent. Convergently parallel lines are modeled by arcs of circles that are tangent to each other at the line at infinity; the tangent lines to these circles coincide there, making an angle of $0$. $\endgroup$
    – Blue
    Jul 15, 2020 at 6:08

2 Answers 2

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More accurately, two distinct lines in a projective plane are never parallel.

This being the case, there must necessarily be an angle between them.

Not necessarily.... why would there be? Angles do not play a role in projective geometry. As Wikipedia mentions:

It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations”

Similarly there is no notion of distance. The thing that takes their place is called the cross-ratio.

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  • $\begingroup$ but if two lines meet, there is an angle between them, I follow what you're saying but surely it would be more accurate to simply say that the angle is unquantifiable (under projective geometry)? $\endgroup$ Jul 15, 2020 at 3:17
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    $\begingroup$ "If two lines meet, there is an angle between then" is only true in Euclidian geometry. Saying "there is an angle, but we can't say nothing about it" is pointless. $\endgroup$ Jul 15, 2020 at 3:31
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    $\begingroup$ @TheIronKnuckle any book on projective geometry. Or start with the wiki $\endgroup$
    – rschwieb
    Jul 15, 2020 at 3:45
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    $\begingroup$ @CélioAugusto well, that is not necessarily true either. Hyperbolic planes can have both metrics and angles. It is not only Euclidean. $\endgroup$
    – rschwieb
    Jul 15, 2020 at 3:47
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    $\begingroup$ @rschwieb My mistake, I should not said "only". $\endgroup$ Jul 16, 2020 at 1:33
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We speak of the shortest geodesic lines in each geometry.There need be no angle between the parallels.

Euclidean

Parallels never meet, or always meet at a point at infinity.

Elliptic

3D models No parallels. Geodesics intersect at a point always. Parallels on a sphere are not geodesics, except the equator.

Hyperbolic Plane models

Poincare models half-plane and disk models. Parallels meet at a point at infinity, on the boundary axis or circle. There are non-intersecting hyper-parallels.

3D model Constant negative Gauss curvature surface. Two parallel sets with no self intersection in each set .Two geodesic parallel asymptotes through each saddle point from either of two sets.

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